algebra word problem

Question

During a research experiment, it was found that the number of bacteria in a culture grew at a rate proportional to its size. At 7:00am there were 6,000 bacteria present in the culture. At noon, the number of bacteria grew to 6,700. How many bacteria will there be at midnight?

Answer 1

If the growth rate is proportional to the population, then it is exponential.  That is the number of bacteria at time t, call it B(t)=B(0)e^kt.  In our case we have B(t)=6000e^kt.  The k is determined by noting that 12 noon, t=5 we have 6700=6000e^5k, and so 5k=ln(6700/6000), or k=.022069611433773. At midnight, t=17 hours we have that

B(17)=6000e^.37518339...=6000×1.45525827...=8731.549... or about 8732

Answer 2

At 7am no of bacteria present.       = 6000

 

At 12 noon no of bacteria present   = 6700

 

In 5 hours increase in bacteria.       = 6700-6000=700

 

In one hour increase in bacteria       = 700/5=140

 

In 12 hours(at mid night) increase in bacteria.      = 140*12=      1680

 

So, no of bacteria present at midnight. = 6700+1680=8380